This continuing project develops new methods in statistics using algebraic techniques, and makes applications of the new methods for biomedical research. Recent work has centered on computational and analytic solutions and simplifications for the EM (Expectation-Maximization) algorithm, using a variety of algebra techniques. These include Jordan algebras, group theory, and group representation theory. The results have been applied to testing and estimation in the presence of missing data and for constrained estimation, given multivariate normal data. Also, already-established and well-known methods have been recently shown, in fact, to depend on some of these algebraic results, further validating our approach. These methods apply to a wide range of statistical problems that seek to analyze biomedical data, including: repeated measures designs, missing data problems, patterned (constrained) covariance estimation, and growth models.